A GEOMETRIC VIEW OF RATIONAL LANDEN TRANSFORMATIONS

Transcription

1 Bull. London Math. Soc. 35 ( C 3 London Mathematical Society DOI:./S A GEOMETRIC VIEW OF RATIONAL LANDEN TRANSFORMATIONS JOHN HUBBARD and VICTOR MOLL Abstract In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.. Introduction The transformation theory of elliptic integrals was initiated by Landen in [6, 7], wherein he proved the invariance of the function under the transformation G(a, b = π/ dθ a cos θ + b sin θ (. a = a + b, b = ab. (. Gauss [4] rediscovered this invariance in the process of calculating the arclength of alemniscate. The limit of the sequence (a n,b n defined by iteration of (. is the celebrated arithmetic-geometric mean AGM(a, b of a and b. The invariance of the elliptic integral (. leads to π = G(a, b. (.3 AGM(a, b General information about the AGM and its applications is given in [3]. A geometric interpretation of the transformation (. is given in [5]. Atransformation analogous to the Gauss Landen map (. has been given in [] for the rational integral U 6 (a,a ; b,b,b = Indeed, the integral U 6 is invariant under the transformation = a a +5a +5a +9 (a + a + 4/3, a ( = a + a +6 (a + a + /3, = b + b + b (a + a + /3, b z 4 + b z + b z 6 + a z 4 + a z dz. (.4 + Received 6 October ; revised July. Mathematics Subject Classification Primary 33E, Secondary 4K.

2 94 john hubbard and victor moll = b (a + + b + b (a + 3, a + a + b + b =. (.5 (a + a + /3 This transformation was obtained by a sequence of elementary changes of variable and the convergence of (a n, b n := ( a (n,a(n,b(n,b(n,b(n was discussed in []: for any initial data (a, b there exists a number L, depending upon the initial condition, such that so that The invariance of U 6 under (.5 shows that (a n, b n (3, 3, L, L, L, (.6 U 6 (a n, b n L π. (.7 U 6 (a, b =L π (.8 as n. Therefore the iteration given above becomes an iterative procedure for evaluating the integral. The main result of [], quoted below, is an extension of (.5 for an even integrand. Theorem.. Define Let R(z =P (z/q(z, with p P (z = b j z (p j and Q(z = j= a j =, for j > p, b j =, for j > p, d p+ j = d = p a j z (p j. (.9 j a p k a j k, for j p, (. k= p a p k, (. k= p c j = a j b p j+k, for j p, (. k= j= and α p (i = i p+ i + k= k + i i ( k +i k d k+i, if i p, p d k, if i =. k= (.3

3 Let and b + i = Q( i/p+/p a + i = [ p i landen transformations 95 α p (i, for i p, (.4 i Q( ( i/p ( ] p k + i (c k + c p k, for i p. i k= Finally, define the polynomials Then p P + (z = b + i z (p i and Q + (z = k= (.5 p a + i z (p i. (.6 k= P (z Q(z dz = P + (z Q + dz. (.7 (z The proofs in [, ] are elementary but lack a proper geometric interpretation. In particular, the proof of (.6 given in [] could not be extended even for degree 8 in view of the formidable algebraic difficulties involved in the arguments given in []. The goal of this paper is to show that the transformation ((.4, (.5 is a particular case of a general construction: the direct image of a meromorphic -form under a rational map. This will allow us to prove an analogue of ((.6, (.8 for the integral b z p + b z p b p U p (a, b := z p + a z p dz. ( In fact, we prove that the sequence x n starting at and defined by x n+ = x + n satisfies (( ( ( p p p x n,,..., ; p where x =(a,...,a p ; b,...,b p ( p L, L = π U p(a, b. ( p ( p L,..., L, p Moreover, the convergence of the iteration is equivalent to the convergence of the initial integral.. The direct image of a -form Let π : X Y be aproper analytic mapping of Riemann surfaces (that is, a finite ramified covering space, and let ϕ be a tensor of any type on X. Then π ϕ is the tensor of the same type on Y,defined as follows. Let U Y be asimply connected subset of Y containing no critical value of π, and let σ,...,σ k : U X be the distinct sections of π. Then the direct image of π ϕ is defined by π ϕ U = k σj ϕ. (. j=

4 96 john hubbard and victor moll This defines π ϕ except at the ramification values of π, where π ϕ may acquire poles even if ϕ is holomorphic. We shall be applying this construction in the case where ϕ is a holomorphic -form, and in this case π ϕ is analytic. Lemma.. If π : X Y is proper and analytic as above, and ϕ is an analytic -form on X, then π ϕ is an analytic -form on Y. Furthermore, for any oriented rectifiable curve γ on Y, we have π ϕ = ϕ. γ π γ Proof. The only problem is to show that π ϕ is holomorphic at the critical values. It is clearly enough to show that the contribution of a neighborhood of a single critical point is holomorphic. Thus we may assume that π(z =w = z m for some m, and that ϕ =(a k z k + a k+ z k+ +...dz, with k. For j =,...,m, set σ j (w =ζ j σ (w, where ζ = e πi/m and σ (w =w /m for some branch of the /m power, for instance the one where the argument is between and π/m. Then { π (z k, if k +is not divisible by m, dz = w (k+ m/m (. dw, if k +is divisible by m. Thus the first term of the power series for ϕ to contribute anything to π ϕ is the term of degree m, and it contributes to the constant term; similarly, the terms of degree m, 3m,... contribute to the terms of degree,,..., all positive powers. This has a useful corollary. Recall that the degree of a meromorphic function is the maximum of the degrees of the numerator and the denominator when the rational function is written in reduced form. Lemma.. If π : is analytic, and ϕ = R(zdz is a meromorphic -form on so that R is a rational function of degree k, then π ϕ can be written as R (zdz, where R is a rational function of degree at most k. Proof. By Lemma., the number of poles of π ϕ is at most equal to the number of poles of ϕ, and their orders cannot increase either. Note. It is quite possible for the degree of π ϕ to be less than the degree of ϕ. This can happen in two ways: we might have poles at two points z and z, such that π(z =π(z, and then the polar parts at these points could cancel. We may also have a pole of order greater then at a critical point, and then the order of the pole at the corresponding critical value might decrease. (In fact, the pole might disappear altogether.

5 landen transformations A particular branched cover We shall be concerned with the specific map π(z =w := z. (3. z This mapping can also be viewed as the Newton map associated to the equation z + =. As such, it has ±i as superattractive fixed points, and π is conjugate to F(z =z via the Möbius transformation M(z =(z + i/(z i; indeed, M π M = F. Let us list some properties of π. Lemma 3.. If ϕ has no poles on, then ϕ = π ϕ. Proof. If ϕ has no poles on (including at infinity, then the integral converges. Since π maps the real axis (including toitself as a double cover, the result follows from Lemma.. Let τ : be the map z z. Then, clearly, π τ = τ π. Call ϕ even if τ ϕ = ϕ, and odd if τ ϕ = ϕ. Note. When ϕ = R(z dz with R a rational function, then ϕ is even if and only if R is odd, and ϕ is odd if and only if R is even, since dz is odd. Lemma 3.. We have the following identities. (a π π ϕ = ϕ + τ ϕ. (b If ϕ is even, then π ϕ =. (c If ϕ is odd, then π ϕ is also odd. Thus we can restrict our attention to odd -forms. Below we calculate π (R(z dz, where R(z is an even rational function. We shall consider only the case when the numerator of R has degree at least less than the denominator, as this avoids a pole at infinity, which would prevent the integral over from converging. The explicit evaluations of the form π ϕ described below were conducted using Mathematica. The corresponding sections are so that for ϕ = Φ(zdz we have σ ± (w =w ± w +, (3. π ϕ = Φ(σ + (w dσ + dw + Φ(σ (w dσ dw. (3.3 The calculations require a symbolic language, since they involve a formidable amount of algebraic manipulation. Example. Let ϕ = b a z + a dz. (3.4

6 98 john hubbard and victor moll Then π ϕ = b (a + a 4a a w dw. (3.5 +(a + a Observe that the new -form can be written as A(a,a π ϕ = b G (a,a w + A dw, (3.6 (a,a where A(a, b and G(a, b are the arithmetic and geometric means of a and b respectively. Example. The form b z + b ϕ = a z 4 + a z dz (3.7 + a is transformed into 8(a b + a b w + (a + a + a (b + b π ϕ = 6a a w 4 + 4(a a +4a a + a a w dw. (3.8 +(a + a + a 4. The convergence of (π n ϕ In this section we present the principal result of the paper. Theorem 4.. Let ϕ be a -form, holomorphic on a neighborhood U of. Then lim (π n ϕ = ( dz ϕ n π +z, where the convergence is uniform on compact subsets of U. Proof. We find it convenient to prove this for the map F(z =z, which is conjugate to π. In that form, the statement to be proved is that if ϕ is analytic in some neighborhood U of the unit circle, then lim (F n ϕ = ( dz n πi z. S ϕ Any such -form ϕ can be developed in a Laurent series ( ϕ = a k z k dz z, k= where k= ( a k + a k ρ k < for some ρ>. Note that a = ϕ. πi S In this form it is very easy to compute F ϕ. Lemma 4.. The mapping F on -forms is given by F ϕ = a k z k dz z. k= Proof. This is what was computed in Equation..

7 landen transformations 99 Thus in the basis of forms z k dz/z, the vector corresponding to k = is an eigenvector with eigenvalue, and the rest of the space is nilpotent: (F m z k dz z = if m is greater than the greatest power of that divides k. This comes close to proving Theorem 4., but this argument does not rule out ( z k k= dz z = dz z( z, which is also fixed under F.Wecannot argue merely in terms of formal Laurent series: convergence must be taken into account. But this is not hard. Consider the region U R defined by /R < z <R, and the space A R of analytic -forms on U R such that We then have ϕ = φ = a + π ϕ n dz a = z = ( k= a k z k dz z ( a k + a k R k <. k= ( a n k + a n k R k k= ( a n k + a n k R n k Rk R n k k= R ϕ. R n This certainly shows that π ϕ a n dz/z tends to, in fact very fast: it superconverges to. 5. Normalization of the integrands In the previous section we produced a map π of -forms ϕ = R(z dz that does not increase the degree and the integral over [, ]. Moreover, we showed that the integrands π n ϕ converge as n tends to infinity. This does not imply the convergence of the coefficients of R, because of possible common factors and cancellations. Here we normalize the rational functions so that π induces a convergent iteration on the coefficients. We shall write the integrands so that their denominators are monic and with constant term equal to. The latter can be achieved by factoring out the constant term, while the former is obtained by a change of variable of the form z λz, with an appropriate λ. Example 3. For rational functions of degree, we obtain b a z + a dz = b (a + a 4a a w dw. (5. +(a + a

8 3 john hubbard and victor moll This is an identity: both sides normalize to b a a Example 4. where The quartic case yields b z + b a z 4 + a z + a dz = The normalization shows that equals (a + a + a / = 8(a b + a b, = (a + a + a (b + b, = 6a a, = 4(a a +4a a + a a, dx x +. (5. w + w4 + w + dw, (5.3 =(a + a + a. (5.4 b a / z + b a / z 4 + a / a a / z + dz (a b + a b w +(b + b a / a / w 4 + [ (a a +4a a + a a a / a / (a + a + a ] w + dw. Naturally, this identity can be verified directly, using Example 5. where dx x 4 +ax + = In the case of degree 6 we obtain b z 4 + b z + b a z 6 + a z 4 + a z + a 3 dz = = 3(a 3 b + a b, x dx x 4 +ax + = π 3/ a +. w4 + w + w6 + w4 + w + dw, (5.5 3 = 8(a b +3a 3 b + a b + a 3 b +3a b + a b, = (a + a + a + a 3 (b + b + b, = 64a a 3, = 6(a a +6a a 3 + a a 3, = 4(a a +4a a + a a +9a a 3 +4a a 3 + a a 3, 3 =(a + a + a + a 3. (5.6 The normalization of (5.5 yields (.5.

9 landen transformations 3 Acknowledgements. The second author acknowledges the partial support of NSF- DMS 7567, Project number This work originated on a field trip to the rain forest in Puerto Rico during SIMU. The authors thank Herbert Medina and Ivelisse Rubio for their hospitality. The SIMU was funded by grants from the National Security Agency (NSA Grant MDA and the National Science Foundation (NSF Grant DMS References. G. Boros and V. Moll, A rational Landen transformation. The case of degree six, Contemp. Math. 5 ( G. Boros and V. Moll, Landen transformation and the integration of rational functions, Math. Comp. 7 ( P. Borwein and J. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, Canad. Math. Soc. Ser. Monogr. Adv. Texts (Wiley, New York, C. F. Gauss, Arithmetische Geometrisches Mittel, Werke vol. 3 (Königliche Gesellschaft der Wissenschaften, Gottingen, (Reprinted by Olms, Hildescheim, D. Grayson, The arithogeometric mean, Arch. Math. 5 ( J. Landen, A disquisition concerning certain fluents, which are assignable by the arcs of the conic sections; wherein are investigated some new and useful theorems for computing such fluents, Philos. Trans. Royal Soc. London 6 ( J. Landen, An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom, Philos. Trans. Royal Soc. London 65 ( H. McKean and V. Moll, Elliptic curves: function theory, geometry, arithmetic (Cambridge University Press, 997. John Hubbard Department of Mathematics Cornell University Ithaca, NY USA jhh8@cornell.edu Victor Moll Department of Mathematics Tulane University New Orleans, LA 78 USA vhm@math.tulane.edu